A First-Order Mean-Field Game on a Bounded Domain with Mixed Boundary Conditions

Entry-exit dynamics are crucial in modeling crowd movement. Here, we present a novel first-order, stationary mean-field game (MFG) model on bounded domains that accurately captures entry-exit dynamics. In our model, the interior dynamics are governed by a standard first-order stationary MFG system: a first-order Hamilton-Jacobi equation coupled with a transport equation. The model incorporates mixed boundary conditions that correspond to an entry region N and an exit region D. A Neumann condition on N prescribes the agent inflow via a non-homogeneous flux term, j (x) ; a no-entry condition on D restricts this boundary region to exit only, preventing inward flow; finally, in D, we prescribe an upper bound on the exit cost combined with a complementary contact-set condition. This contact-set condition identifies boundary points where the value function attains the exit cost (contact points) versus points where the non-penetration condition prevents artificial inflows (non-contact points). However, as our examples show, contact does not necessarily imply that exit occurs. This mixed approach overcomes the limitations of classical Dirichlet conditions, which can artificially force boundary points to act as both entry and exit sites. We analyze the system using a variational formulation, applying the direct method of calculus of variations to establish the existence of solutions under minimal regularity assumptions. Furthermore, we prove the uniqueness of the gradient of the value function (particularly in regions with positive agent density) and the uniqueness of the density function. Several examples, including cases in one and two dimensions, illustrate first-order MFG phenomena such as the formation of empty regions (where agent density vanishes) and the proper assignment of entry and exit roles. These results establish a rigorous mathematical foundation for modeling realistic entry-exit scenarios.